I am very confused on how to graph Polynomial Functions and how to determine the extra stuff my HW is asking.
a. Graph each function by making a table of values.
b. Determine consecutive values of x between which each real zero is located.
c. Estimate the x-coordinates at which the relative maxima and relative minima occur.
problem-
f(x)=x(cubed)-2x(squared)+ 6
Just take the first one I have to do and please help me understand how to do this and what all this means!! Thanks!
-JAKE
Personally, I'd do those backwards from c, and I was taught to sketch graphs using a different method, so maybe someone else will be more in tune with your text. Anyway, following these steps:
a) Start with a table of values. This function is most interesting in x [-3, 3], so create a table of values of y given x:
f(-2) =(-2)^2-a(-2)^2+6
f(-1) =
f(0) =
f(1) =
f(2) =
f(3) =
Extend a few more values between or beyond, to taste.
b) Where does the function cross the x-axis? That is, where does it cross from negative to positive or vice-versa? If it's positive at 2 but negative at three you know you have (at least) one real root between.
c. The actual answer to this is to differentiate the function, and find where the derivative is zero.
f'(x)=3x^2-4x at zero,
3x^2 = 4x, which is true at x=0 and at x=4/3, so these points are going to be especially interesting.
Sure Jake, I'd be happy to help you understand how to graph polynomial functions and determine the extra information your homework is asking for.
a. To graph a polynomial function, we start with a table of values. We choose different x-values and calculate the corresponding y-values using the given function. For your problem, f(x) = x^3 - 2x^2 + 6, we can start with a table like this:
x | f(x)
----------------
-3 | ?
-2 | ?
-1 | ?
0 | ?
1 | ?
2 | ?
3 | ?
To find the y-values, simply substitute each x-value into the given function and calculate the result:
For x = -3: f(-3) = (-3)^3 - 2(-3)^2 + 6 = -9 + 18 + 6 = 15
Fill in the table with the remaining values.
b. To find the consecutive values of x between which each real zero is located, we need to determine the x-values for which f(x) equals zero. In other words, we are looking for solutions to the equation f(x) = 0. In this case, we have the equation x^3 - 2x^2 + 6 = 0.
There are different methods to solve this equation, such as factoring, using the quadratic formula, or using numerical methods like Newton's method. One way to start is by factoring out common terms, if possible. Unfortunately, this equation can't be factored easily.
We can use a graphing calculator or software to visualize the graph and locate the x-values where the curve intersects the x-axis. These are the consecutive values of x between which each real zero is located. In the case of this function, it appears that there is only one real zero, somewhere between x = 2 and x = 3.
c. To estimate the x-coordinates at which the relative maxima and relative minima occur, we need to find the highest and lowest points on the graph, respectively. Relative maxima and minima are points where the slope changes from positive to negative or vice versa.
One way to find these points is by taking the derivative of the function, which gives us the slope of the curve at any point. For your function, f(x) = x^3 - 2x^2 + 6, the derivative is f'(x) = 3x^2 - 4x.
To find the x-coordinates for relative maxima and minima, we set the derivative equal to zero and solve for x. So, f'(x) = 0:
3x^2 - 4x = 0
Factoring, we get:
x(3x - 4) = 0
This equation has two solutions: x = 0 and x = 4/3. These are the x-coordinates where relative maxima and minima might occur. To determine if they are relative maxima or minima, we can use the second derivative test. By taking the second derivative and evaluating it at the critical points, we can determine whether the function is concave up or concave down at those points.
By following these steps, you should be able to graph the function, determine the consecutive values of x between which the real zero is located, and estimate the x-coordinates of the relative maxima and minima. Let me know if you have any more questions or if there's anything else I can help you with!